Except for the claim that the model structures are symmetric monoidal, all assertions follow from [1]}.
The model structures for sheaves of \(\infty \) -groupoids are symmetric monoidal by [1]}.
To see that \({H}_\infty ^{p/i\, I}\)  is symmetric monoidal, observe that the objects \({{Y}}_c\) , \(c \in {{C}\mathrm {art}}\) , form a set of homotopy generators for \({H}_\infty ^{p/i}\) .
Let \(F \in {H}_\infty ^{p/i\, I}\)  be a local object, and consider the internal hom object \(F^{{{Y}}_d}\)  for any \(d \in {{C}\mathrm {art}}\) .
For any of the morphisms \({{Y}}_c \times {{Y}}_\mathbb {R}\rightarrow {{Y}}_c\)  in \(I\) , the internal hom adjunction yields a commutative diagram of simplicially enriched hom spaces
\(\begin{tikzcd}[row sep=1cm]\underline{{H}}_\infty ({{Y}}_c, F^{{{Y}}_d}) [r] [d, "\cong "^{\prime }]& \underline{{H}}_\infty ({{Y}}_c \times {{Y}}_\mathbb {R}, F^{{{Y}}_d}) [d, "\cong "]\\\underline{{H}}_\infty ({{Y}}_{c \times d}, F) [r]& \underline{{H}}_\infty ({{Y}}_{c \times d} \times {{Y}}_\mathbb {R}, F)\end{tikzcd}\) 
